298 ANALYTICAL MECHANICS
Putting equation (II) in the form
and integrating we obtain
sn"— = u+d,
or x = — sin (ut + 5)
where <5 is the constant of integration and a = ~ •
227. Displacement. — The distance, x, of the particle from -the fixed point is called the displacement.
228. Amplitude. — The maximum displacement is called the amplitude. It is evident from equation (III) that the amplitude equals a.
229. Phase. — The particle is said to be in the same phase at two different instants, if the displacement and the velocity at the one instant equal, respectively, the displacement and the velocity at the other instant.
230. Period. — The time which elapses between two successive instants at which the particle is in the same phase is called the period of the motion. In order to find the period we will make use of the definition of a periodic function.* It is evident from equation (III) that x is a periodic function of t\ therefore we can write
x = a sin [at + d]
* If any variable x is a periodic function of any other variable t and if the dependence of x on t is given by the relation x » / (t), then the function satis* fies the following condition: